Example Data Table
Use these examples to test the calculator quickly.
| Polynomial A |
Polynomial B |
Expected GCD |
Reason |
x^4 - 1 |
x^3 - x |
x^2 - 1 |
Both contain (x - 1)(x + 1). |
x^3 - 3x^2 + 2x |
x^2 - x |
x^2 - x |
The second polynomial divides the first. |
2x^3 + 4x^2 + 2x |
x^2 + 2x + 1 |
x + 1 |
The monic common factor is returned. |
x^2 + 1 |
x + 3 |
1 |
No nonconstant common polynomial factor exists. |
Formula Used
The calculator uses the polynomial Euclidean algorithm.
For two polynomials A(x) and B(x), divide:
A(x) = B(x)Q(x) + R(x).
The remainder must satisfy degree(R) < degree(B).
Then repeat the process with B(x) and R(x).
The final nonzero remainder is the polynomial GCD.
With monic normalization, it is divided by its leading coefficient.
The recursive identity is:
gcd(A, B) = gcd(B, R).
How to Use This Calculator
- Enter the first polynomial in the Polynomial A field.
- Enter the second polynomial in the Polynomial B field.
- Choose the variable, such as
x, t, or y.
- Select a normalization option for the final GCD.
- Adjust tolerance if decimal coefficients create tiny noise.
- Set graph limits and sample points if needed.
- Click the calculate button.
- Review the result, Euclidean table, graph, CSV, and PDF options.
Polynomial GCD Basics
A polynomial greatest common divisor is the largest polynomial factor shared by two inputs.
It works much like a number GCD.
The difference is that each step handles powers, coefficients, and remainders.
This calculator uses the Euclidean algorithm.
It divides the first polynomial by the second.
Then it divides the second polynomial by the remainder.
The cycle continues until the remainder becomes zero.
Why The Method Matters
The polynomial GCD helps simplify rational expressions.
It also supports factor checking, algebra proofs, coding theory, and symbolic computation.
When two expressions share a repeated factor, the GCD exposes it.
This is useful before canceling terms.
It is also useful when testing whether two equations have a common root pattern.
How The Output Helps
The result panel shows the cleaned input polynomials first.
It then lists degrees, leading coefficients, the final GCD, and reduced cofactors.
The Euclidean table shows each quotient and remainder.
This makes the work easy to audit.
You can compare every division step with a manual solution.
The graph is not a proof.
Yet it helps visualize shared behavior near roots.
Normalization And Precision
Polynomial GCDs are not unique in raw form.
Any nonzero constant multiple is also a valid divisor.
For that reason, most algebra systems return a monic GCD.
A monic polynomial has leading coefficient one.
This page also lets you keep the raw result, or only force a positive leading coefficient.
Decimal coefficients may create tiny rounding noise.
The tolerance setting removes very small values from the final expression.
Best Practice
Use one variable at a time.
Write powers with the caret symbol.
Fractions like 3/2 are allowed.
Decimals are allowed too.
For exact classroom work, prefer integer or fractional coefficients.
After calculation, download the CSV for records.
Use the PDF button when you need a printable report.
The example table gives quick patterns for testing.
If a result is one, the inputs have no nonconstant common factor.
If the GCD has degree one or more, both inputs share a real or complex algebraic factor.
Always check the quotient rows.
They reveal division mistakes and confirm that the final remainder sequence is consistent.
FAQs
1. What is a polynomial GCD?
It is the greatest polynomial factor shared by two polynomials. It may be a constant, linear factor, or higher degree expression.
2. Why does the calculator return a monic result?
Polynomial GCDs can differ by nonzero constants. A monic result keeps the answer standard by making the leading coefficient equal to one.
3. Can I use fractions?
Yes. You can enter fractions like 3/2x^2 or -5/4x. Use one slash per coefficient and avoid spaces inside fractions.
4. Can I use decimals?
Yes. Decimals are accepted. For cleaner symbolic answers, fractions or integers usually produce more reliable algebraic output.
5. What does GCD equal to one mean?
It means the two polynomials have no nonconstant common polynomial factor under the selected precision and normalization settings.
6. Why is there a tolerance option?
Floating calculations can create very small leftover coefficients. The tolerance treats tiny values as zero to keep the output readable.
7. Does the graph prove the GCD?
No. The graph is only a visual aid. The Euclidean division table is the real algebraic evidence for the result.
8. Which variable names are supported?
Use one letter as the variable. Examples include x, y, z, or t. Do not mix different variables in one calculation.